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WASHINGTON - However the debt limit showdown ends, one thing is clear: under pressure from congressional Republicans, President Obama has moved rightward on budget policy, deepening a rift within his party heading into the next election.

Entering into a campaign that is shaping up as an epic clash concerning the parties’ divergent views on the size and role of the federal government, Republicans have changed the terms of the national debate.

Obama, seeking to appeal to the broad swath of independent voters, has adopted the Republicans’ language and in some cases their policies while signaling a willingness to break with liberals on some issues.

That has some progressive members of Congress and liberal groups arguing that by not fighting for more stimulus spending, Obama could be left with an economy still producing so few jobs by Election Day that his reelection could be threatened.

Not only would independents be turned off, but Obama risks alienating Democratic voters already disappointed by his escalation of the war in Afghanistan, and by his failure to close the Guantanamo Bay prison, end the Bush-era tax cuts, and enact a government-run health insurance system.

“The activist liberal base will support Obama because they’re terrified of the right wing,’’ said Robert L. Borosage, codirector of the liberal group Campaign for America’s Future.

He added: “But I believe that the voting base of the Democratic Party - young people, single women, African-Americans, Latinos - are going to be so discouraged by this economy and so dismayed unless the president starts to champion a jobs program and take on the Republican Congress that the ability of labor to turn out its vote, the ability of activists to mobilize that vote, is going to be dramatically reduced.’’

Obama’s efforts at compromise have pushed the debate firmly to the right. In his failed effort to negotiate a broad budget deal with Republicans, Obama proposed far more in reductions for future years’ spending, including from Medicare, Medicaid, and Social Security, than he did new revenue from the wealthy and corporations.

He proposed fewer cuts in military spending and more in health care programs than a bipartisan Senate group that includes one of that chamber’s most conservative Republicans.

To win congressional approval of the essential increase in the nation’s borrowing ceiling, Obama sought more in deficit reduction than Republicans did, with fewer changes to the entitlement benefit programs, because he was willing to raise additional revenue starting in 2013 and they were not.

And despite unemployment at its highest level in decades, Obama has not fought this year for a big jobs program with billions of dollars for public-works projects, which liberals in his party have sought.

Instead, he wants to extend a temporary payroll tax cut for everyone, since Republicans will support tax cuts, despite studies showing that spending programs are generally the more effective stimulus.

Even before last November’s election gave the Republicans control of the House, Obama had said he would pivot to deficit reduction after two years of stimulus measures intended first to rescue the economy and then to spur a recovery from the near-collapse of the financial system.

In his annual budget proposal in January, Obama declined to suggest a plan along the lines proposed by a majority of his bipartisan fiscal commission, which in December recommended $4 trillion in savings over 10 years through cuts in military and domestic programs, including Medicare and Medicaid, and an overhaul of the tax code to lower rates while also raising more revenue.

Even though Obama was widely criticized, administration officials said at the time that to have embraced that approach then would have put him too far to the right - where he ultimately wanted to end up in any compromise with Republicans, not where he wanted to start.

But by this month, in private talks with House Speaker John A. Boehner to find a deficit-reduction compromise, Obama tentatively agreed to a plan that was farther to the right than that of the majority of the fiscal commission and a bipartisan group of senators, the so-called Gang of Six. It also included a slow rise in the Medicare eligibility age to 67 from 65, and, after 2015, a change in the formula for Social Security cost-of-living adjustments long sought by economists.

“He’s accommodated himself to the new reality in Washington,’’ said Tom Davis, a former House Republican leader from Virginia. “That’s what leaders do.’’

The president’s advisers express confidence that voters will reward Obama either for winning a bipartisan deal, if that were to happen, or for having a more balanced approach that does not remake Medicare and Medicaid as Republicans have proposed and asks for more revenue from the wealthy.